3.893 \(\int \frac{x^7}{\left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\sqrt{1-x^4}}{2}+\frac{1}{2 \sqrt{1-x^4}} \]

[Out]

1/(2*Sqrt[1 - x^4]) + Sqrt[1 - x^4]/2

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Rubi [A]  time = 0.0393266, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\sqrt{1-x^4}}{2}+\frac{1}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(1 - x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 - x^4]) + Sqrt[1 - x^4]/2

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Rubi in Sympy [A]  time = 4.54226, size = 20, normalized size = 0.65 \[ \frac{\sqrt{- x^{4} + 1}}{2} + \frac{1}{2 \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(-x**4+1)**(3/2),x)

[Out]

sqrt(-x**4 + 1)/2 + 1/(2*sqrt(-x**4 + 1))

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Mathematica [A]  time = 0.0110737, size = 22, normalized size = 0.71 \[ \frac{2-x^4}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(1 - x^4)^(3/2),x]

[Out]

(2 - x^4)/(2*Sqrt[1 - x^4])

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Maple [A]  time = 0.006, size = 28, normalized size = 0.9 \[{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ({x}^{4}-2 \right ) }{2} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(-x^4+1)^(3/2),x)

[Out]

1/2*(-1+x)*(1+x)*(x^2+1)*(x^4-2)/(-x^4+1)^(3/2)

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Maxima [A]  time = 1.4395, size = 31, normalized size = 1. \[ \frac{1}{2} \, \sqrt{-x^{4} + 1} + \frac{1}{2 \, \sqrt{-x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

1/2*sqrt(-x^4 + 1) + 1/2/sqrt(-x^4 + 1)

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Fricas [A]  time = 0.284799, size = 22, normalized size = 0.71 \[ -\frac{x^{4} - 2}{2 \, \sqrt{-x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

-1/2*(x^4 - 2)/sqrt(-x^4 + 1)

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Sympy [A]  time = 2.3447, size = 22, normalized size = 0.71 \[ - \frac{x^{4}}{2 \sqrt{- x^{4} + 1}} + \frac{1}{\sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(-x**4+1)**(3/2),x)

[Out]

-x**4/(2*sqrt(-x**4 + 1)) + 1/sqrt(-x**4 + 1)

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GIAC/XCAS [A]  time = 0.211369, size = 31, normalized size = 1. \[ \frac{1}{2} \, \sqrt{-x^{4} + 1} + \frac{1}{2 \, \sqrt{-x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

1/2*sqrt(-x^4 + 1) + 1/2/sqrt(-x^4 + 1)